Efficient harmonic generation and frequency conversion in multi-mode cavities

ABSTRACT

A doubly-resonant cavity structure includes at least one cavity structures so as to allow total frequency conversion for second or third-harmonic generation using χ (2)  and χ (3)  nonlinearities between the at least one cavity structures. The total frequency conversion is efficiently optimized by determining a critical power allowing for such total frequency conversion to occur depending on the cavity parameters of the at least one cavity structures.

PRIORITY INFORMATION

This application claims priority from provisional application Ser. No.60/889,566 filed Feb. 13, 2007, which is incorporated herein byreference in its entirety.

This invention was made with government support awarded by the NationalScience Foundation under Contract No. DMR-9400334. The government hascertain rights in the invention.

BACKGROUND OF THE INVENTION

The invention relates to the field of coupled cavity structures, and inparticular to an efficient harmonic generation and frequency conversionscheme in multi-mode cavity structures.

Nonlinear frequency conversion has been commonly realized in the contextof waveguides, or even for free propagation in the nonlinear materials,in which light at one frequency co-propagates with the generated lightat the harmonic frequency. A phase-matching condition between the twofrequencies must be satisfied in this case in order to obtain efficientconversion. Moreover, as the input power is increased, the frequencyconversion eventually saturates due to competition between up and downconversion. Previous experimental and theoretical work onsecond-harmonic generation in cavities has largely focused on cavitieswith a single resonant mode at the pump frequency. Such structuresrequire much higher powers than our proposed doubly-resonant cavity,however, approaching one Watt and/or requiring amplification within thecavity.

Second-harmonic generation in a doubly resonant cavity, with a resonanceat both the pump and harmonic frequency, have previously been analyzedonly in the limit where nonlinear down-conversion can be neglected.Previous work on third-harmonic generation in cavities, similarly,considered only singly resonant cavities; moreover, past work focused onthe case of χ⁽²⁾ materials where 3ω is generated by cascading twononlinear processes (harmonic generation and frequency summing).Furthermore, the previous theoretical work, with a few exception,focused on one-dimensional Fabry-Perot cavity geometries, in which theproblem of obtaining cavity modes with the correct frequency ratio wasposed as a problem of phase-matching, and addressed by methods such asusing off-normal beams.

SUMMARY OF THE INVENTION

According to one aspect of the invention, there is provided adoubly-resonant cavity structure. The doubly-resonant cavity structureincludes at least one cavity structures so as to allow total frequencyconversion for second or third-harmonic generation using χ⁽²⁾ and χ⁽³⁾nonlinearities between the at least one cavity structures. The totalfrequency conversion is efficiently optimized by determining a criticalpower allowing for such total frequency conversion to occur depending onthe cavity parameters of the at least one cavity structures.

According to another aspect of the invention, there is provided a methodof performing total frequency conversion in a doubly-resonant cavitystructure. The method includes providing at least one cavity structuresso as to allow the total frequency conversion for second orthird-harmonic generation using χ⁽²⁾ and χ⁽³⁾ nonlinearities between theat least one cavity structures. Also, the method includes determiningthe cavity parameter of the at least one cavity structures. In addition,the method includes determining a critical power to efficientlyoptimized the total frequency conversion using the cavity parameters ofthe at least one cavity structures. Furthermore, the method includesapplying the critical power so as to allow total frequency conversionbetween the at least one cavity structures to occur.

According to another aspect of the invention, there is provided methodof forming a doubly-resonant cavity structure. The method includesforming at least two cavity structures so as to allow total frequencyconversion for second or third-harmonic generation using χ⁽²⁾ and χ⁽³⁾nonlinearities between the at least two cavity structures. Moreover, themethod includes determining the cavity parameters of the at least twocavity structures so as to determine the critical power needed toperform the total frequency conversion.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1B are schematics diagrams illustrating waveguide-cavitysystems used by the invention;

FIG. 2 is a graph illustrating a Log-log plot of |s³⁻|²/|s₁₊|² vs.n₂|s₁₊|² for the coupled-mode theory and FDTD;

FIG. 3 is a graph illustrating first and second harmonic efficiency;

FIG. 4 is a graph illustrating first and third harmonic efficiency;

FIGS. 5A-5B are schematic diagrams illustrating a ring resonator withuniform waveguide coupling and asymmetrical waveguide coupling;

FIG. 6 is schematic diagram illustrating a possible harmonic conversionsystem formed in accordance with the invention; and

FIG. 7 is schematic diagram illustrating a 3d photonic crystal (PhC)cavity created by adding a defect in a rod-layer of a (111) fcc latticeof dielectric rods.

DETAILED DESCRIPTION OF THE INVENTION

The invention permits the generals conditions for 100% frequencyconversion in any doubly resonant nonlinear cavity to occur, for bothsecond- and third-harmonic generation via χ⁽²⁾ and χ⁽³⁾ nonlinearities.Conversion efficiency is optimized for a certain “critical” powerdepending on the cavity parameters, and assuming reasonable parametersone can predict 100% conversion using milliwatts of power or less. Theseresults follow from a general coupled-mode theory framework that isderived for harmonic generation in cavities, and which is verified bydirect finite-difference time-domain (FDTD) simulations of the nonlinearMaxwell equations. Explicit formulas for the nonlinear couplingcoefficients are derived in terms of the linear cavity modes, which canbe used to design and evaluate cavities in arbitrary geometries. Theeffect of linear and nonlinear losses is also analyzed

Frequency conversion in a doubly-resonant cavity, as we shall derive,has three fundamental differences from this familiar case of propagatingmodes. First, light in a cavity can be much more intense for the sameinput power, because of the spatial (modal volume V) and temporal(lifetime Q) confinement. The invention shows that this enhancessecond-harmonic (χ⁽²⁾) conversion by a factor of Q³/V and enhancesthird-harmonic (χ⁽³⁾) conversion by a factor of Q²/V. Second, thephase-matching condition is replaced by the condition that the cavitysupport two modes of the requisite frequencies, the frequencies can bedesigned by tuning any of a number of cavity parameters. Third, thefrequency conversion no longer saturates—instead, it peaks at 100%, withproper design, for a certain critical input power satisfying a resonantcondition, and goes to zero if the power is either too small or toolarge.

FIG. 1A shows a schematic diagram of waveguide-cavity system 2 used bythe invention. Input light from a waveguide 4 at one frequency with anamplitude s₁₊ is coupled to a cavity 6 having a cavity mode with anamplitude a₁, converted to a cavity mode at another frequency with anamplitude a₂ by a nonlinear process, and radiated back into thewaveguide 4 with an amplitude s²⁻. Reflections at the first frequency(s¹⁻) can also occur.

FIG. 1B shows a two-mode nonlinear cavity coupled to an input/outputchannel, and in particular a Fabry-Perot cavity 10 between twoquarter-wave stacks 12, where the stack has fewer layers on one side sothat light can enter/escape. For a nonlinear effect, one can considerspecifically a χ^((l)) nonlinearity, corresponding essentially to ashift in the refractive index proportional to the nonlinearsusceptibility χ^((l)) multiplied by electric field E to the (l−1)^(th)power. Most commonly, one would have either a χ⁽²⁾ or χ⁽³⁾ (Kerr)effect. Such a nonlinearity results in harmonic generation: light withfrequency ω is coupled to light with frequency lω. Therefore, if wedesign the cavity so that it supports two modes, one at ω and one at lω,then input power at ω can be converted, at least partially, to outputpower at lω.

In order to achieve 100% conversion from ω to lω inside a cavity, it isnot always sufficient to simply increase the input power in order toincrease the rate of nonlinear transfer. Putting aside the question ofat what point material breakdown will occur, increasing the input powerfor a ω⁽³⁾ medium (or any odd l) also causes a shift in the resonantfrequency (self-phase modulation) that, unchecked, will prevent 100%conversion by making the frequency ratio ≠l. To address this mismatch,one can use two materials with opposite-sign χ^((l)) to cancel thefrequency-shifting effect; it may also be possible to pre-shift thecavity resonant frequency to correct for the nonlinear shift. On theother hand, a χ⁽²⁾ medium has no self-phase modulation, and so in thiscase it is sufficient to increase the input power until 100% frequencyconversion is reached. Regarding material breakdown, we show that it issufficient to use modes with a large quality factor (lifetime) Q so thata slow conversion due to a weak nonlinear effect has enough time tooccur.

Nonlinear frequency conversion has previously been studied with bothdirect simulation and by semi-analytic perturbative methods. Inparticular, the most common perturbative approach is known as“coupled-wave” or “coupled-mode” theory (CMT): essentially, one writes aset of ordinary differential equations for the amplitudes of the linearmodes, in which these amplitudes are weakly coupled by the nonlinearity.The most common variation of this CMT approach is for waveguides, inwhich the only degrees of freedom are two (or more) coupled waveguidemodes, as described in. In this case, the problem is modified by thefact that the modes are continuously “pumped” by an external input, andthe quantity of interest is the power radiated to an external output.

There are two approaches to developing a CMT for such a problem. First,in the “temporal” CMT, the most general possible CMT is derived fromfundamental principles such as conservation of energy and reciprocity,parameterized by a few unknown frequencies and coupling factors thatreflect the specific geometry. Second, one can apply perturbativeexpansions directly to Maxwell's equations to derive explicitexpressions for the coupling factors. Although both approaches have beensuccessfully employed to describe various nonlinear phenomena, frequencyconversion in doubly-resonant cavities does not seem to have been fullyaddressed. In the following, both approaches are applied to derive boththe most general CMT and also the specific coupling factors for l=2, 3.

The couple-mode equations are derived describing the interaction oflight in a multi-mode cavity filled with nonlinear material and coupledto input/output ports, from which light can couple in (s₊) and out (s⁻)of the cavity. The schematic illustration of the system is shown in FIG.1A. Specifically, the formalism is adapted to handle nonlinearly coupledmodes with frequencies ω_(k), which is parameterized as follows.

The time-dependent complex amplitude of the kth mode is denoted bya_(k), normalized so that |a_(k)|² is the electromagnetic energy storedin this mode. The time-dependent amplitude of the incoming (+) oroutgoing (−) wave is denoted by s_(±), normalized so that |s_(±)|² isthe power. More precisely, s_(±)(t) is normalized so that its Fouriertransform |s^(□) _(±)(ω)|² is the power at ω. By itself, a linear cavitymode decaying with a lifetime τ_(k) would be described byda_(k)/dt=(iω_(k)−1/τ_(k))a_(k). The decay rate 1/τ_(k) can bedecomposed into 1/τ_(k)=1/τ_(e,k)+1/τ_(s,k) where 1/τ_(e,k) is the“external” loss rate (absorption etc.) and 1/τ_(s,k) is the decay rateinto s⁻. When the weak coupling (ω_(k)τ_(k)>>1) to s_(±) is included,energy conservation and similar fundamental constraints lead toequations of the form:

$\begin{matrix}{\frac{\mathbb{d}a_{k}}{\mathbb{d}t} = {{\left( {{\mathbb{i}\omega}_{k} - \frac{1}{\tau_{k}}} \right)a_{k}} + \sqrt{2\tau_{s,k}s_{+}}}} & (1) \\{s_{-} = {{- s_{+}} + {\sqrt{2}\tau_{s,k}a_{k}}}} & (2)\end{matrix}$

This can be generalized to incorporate multiple input/output ports,direct coupling between the ports, and so on. The only unknownparameters in this model are then the frequencies ω_(k) and the decayrates 1/τ_(k), which can be determined by any numerical method to solvefor the cavity modes (e.g. FDTD, below). Instead of τ_(k), one commonlyuses the quality factor Q_(k)=ω_(k)τ_(k)/2.

Nonlinearity modifies this picture with two new amplitude-dependenteffects: a shift in the frequency (and decay rate) of the cavity, and acoupling of one cavity mode to another. The nonlinear effects areneglected on the input/output ports, under the assumption that intensefields are only present in the cavity (due to spatial and temporalconfinement). Two standard assumptions are made of nonlinear systems.First, that the nonlinearities are weak, in the sense that we canneglect terms of order (χ^((l)))² or higher; this is true in practicebecause nonlinear index shifts are always under 1% lest materialbreakdown occur. Second, we make the rotating wave approximation: sincethe coupling is weak, we only include terms for a_(k) that havefrequency near ω_(k). In particular, we suppose that ω_(k)≈kω₁, so thatω_(k) is the kth harmonic. Then, for a χ^(x(2)) nonlinearity with twomodes ω₁ and its second harmonic ω₂, the coupled-mode equations musttake the form:

$\begin{matrix}{\frac{\mathbb{d}a_{1}}{\mathbb{d}t} = {{\left( {{\mathbb{i}\omega}_{1} - \frac{1}{\tau_{1}}} \right)a_{1}} - {{\mathbb{i}\omega}_{1}\beta_{1}a_{1}^{*}a_{2}} + {\sqrt{\frac{2}{\tau_{1,s}}}s_{+}}}} & (3) \\{\frac{\mathbb{d}a_{2}}{\mathbb{d}t} = {{\left( {{\mathbb{i}\omega}_{2} - \frac{1}{\tau_{2}}} \right)a_{2}} - {{\mathbb{i}}\;\omega_{2}\beta_{2}a_{1}^{2}} + {\sqrt{\frac{2}{\tau_{2,s}}}s_{+}}}} & (4)\end{matrix}$

Similarly, for a χ⁽³⁾ nonlinearity with two modes ω₁ and its thirdharmonic ω₃, the coupled-mode equations must take the form:

$\begin{matrix}{\frac{\mathbb{d}a_{1}}{\mathbb{d}t} = {{\left( {{{\mathbb{i}}\;{\omega_{1}\left( {1 - {\alpha_{11}{a_{1}}^{2}} - {\alpha_{13}{a_{3}}^{2}}} \right)}} - \frac{1}{\tau_{1}}} \right)a_{1}} - {{\mathbb{i}}\;\omega_{1}{\beta_{1}\left( {a_{*},1} \right)}^{2}a_{3}} + {\sqrt{\frac{2}{\tau_{1,s}}}s_{+}}}} & (5) \\{{\frac{\mathbb{d}a_{3}}{\mathbb{d}t} = {{\left( {{{\mathbb{i}}\;{\omega_{3}\left( {1 - {\alpha_{33}{a_{3}}^{2}} - {\alpha_{31}{a_{1}}^{2}}} \right)}} - \frac{1}{\tau_{3}}} \right)a_{3}} - {{\mathbb{i}}\;\omega_{3}\beta_{3}a_{3}}}},{1 + {\sqrt{\frac{2}{\tau_{3,s}}}s_{+}}}} & (6)\end{matrix}$

In equations 3-6, one sees two kinds of terms. The first arefrequency-shifting terms, with coefficients α_(ij), dependent on one ofthe field amplitudes. For χ⁽³⁾, this effect is known as self-phase andcross-phase modulation, which is absent for χ⁽²⁾ (under the first-orderrotating-wave approximation). The second kind of term transfers energybetween the modes, with coupling coefficients β_(i), corresponding tofour-wave mixing for χ⁽³⁾. Furthermore, one can constrain the couplingterms β_(i) by energy conservation:

${\frac{\mathbb{d}}{\mathbb{d}t}\left( {{a_{1}}^{2} + {a_{2}}^{2}} \right)} = 0.$For χ⁽²⁾, the constraint that follows is: ω₁β₁=ω₂β*₂; for χ⁽³⁾, theconstraint is ω₁β₁=ω₃β*₃.

The general process for construction of these coupled-mode equations isas follows. The underlying nonlinearity must depend on the physical,real part of the fields, corresponding to (a_(k)+a*_(k))/2. It thenfollows that the χ^((l)) term will have l powers of this real part,giving various product terms like a*₁a₂ (for χ⁽²⁾) and a*₁a₁a₁ (forχ⁽³⁾). Most of these terms, however, can be eliminated by therotating-wave approximation. In particular, we assume that each a_(k)term is proportional to e^(kiω) multiplied by a slowly varying envelope,and we discard any product term whose total frequency differs from kωfor the da_(k)/dt equation. Thus, a term like a*₁a₃a₃ would beproportional to e^(5iω), and would only appear in a da₅/dt equation.

At this point, the equations are already useful in order to reason aboutwhat types of qualitative behaviors are possible in general. In fact,they are not even specific to electromagnetism and would also apply toother situations such as acoustic resonators. However, in order to makequantitative predictions, one needs to know the nonlinear coefficientsα_(ij) and β_(i) (as well as the linear frequencies and decay rates).The evaluation of these coefficients requires a more detailed analysisof Maxwell's equations as described below.

When a dielectric structure is perturbed by a small δ∈, a well-knownresult of perturbation theory states that the corresponding change δω inan eigenfrequency ω is, to first order:

$\begin{matrix}{\frac{\delta\;\omega}{\omega} = {{{- \frac{1}{2}}\frac{\int{{\mathbb{d}^{3}x}\;{\delta ɛ}{E}^{2}}}{\int{{\mathbb{d}^{3}x}\; ɛ{E}^{2}}}} = {{- \frac{1}{2}}\frac{\int{{{\mathbb{d}^{3}{xE}^{*}} \cdot \delta}\; P}}{\int{{\mathbb{d}^{3}x}\; ɛ{E}^{2}}}}}} & (7)\end{matrix}$where E is the unperturbed electric field and δP=δ∈E is the change inpolarization density due to δ∈. In fact, Eq. 7 is general enough to beused with any δP, including the polarization that arises from anonlinear susceptibility. In particular, we can use it to obtain thecoupling coefficients of the CMT.

To do so, one first computes the nonlinear first-order frequencyperturbation due to the total field E present from all of the modes.Once the frequency perturbations δω_(k) are known, one can re-introducethese into the coupled-mode theory by simply setting ω_(k)→ω_(k)+δω_(k)in Eq. 1. By comparison with Eqs. 3-6, the α and β coefficients can thenbe identified.

First a χ⁽²⁾ nonlinearity case is considered, with P given by

${P_{i} = {\sum\limits_{ijk}\;{ɛ\;\chi_{ijk}^{(2)}E_{j}E_{k}}}},$in a cavity with two modes E₁ and E₂. As before, the modes are requiredto oscillate with frequency ω₁ and ω₂≈2ω₁, respectively. TakingE=Re[E₁e^(iω)1^(t)+E₂e^(iω)2^(t)] and using the rotating-waveapproximation, one can separate the contribution of δP to each δω_(k),to obtain the following frequency perturbations:

$\begin{matrix}{{\frac{\delta\;\omega_{1}}{\omega_{1}} = {{- \frac{1}{4}}\frac{\int{{\mathbb{d}^{3}x} \times {\sum\limits_{ijk}\;{ɛ\;{\chi_{ijk}^{(2)}\left\lbrack {E_{1i}^{*}\left( {{E_{2j}E_{1k}^{*}} + {E_{1j}^{*}E_{2k}}} \right)} \right\rbrack}}}}}{\int{{\mathbb{d}^{3}x}\; ɛ{E_{1}}^{2}}}}},} & (8) \\{\frac{\delta\;\omega_{2}}{\omega_{2}} = {\frac{- 1}{4}\frac{\int{\mathbb{d}^{3}{\times {\sum\limits_{ijk}\;{ɛ\;\chi_{ijk}^{(2)}E_{2i}^{*}E_{1j}E_{1k}}}}}}{\int{{\mathbb{d}^{3}x}\; ɛ{E_{2}}2}}}} & (9)\end{matrix}$

Similarly, for a centro-symmetric ω⁽³⁾ medium, P is given byP=∈χ⁽³⁾|E|²E, with E=Re[E₁e^(iω) ¹ ^(t)+E₃e^(iω) ³ ^(t)]. One can obtainthe following frequency perturbations:

$\begin{matrix}{\frac{\delta\;\omega_{1}}{\omega_{1}} = {{- \frac{1}{8}}\begin{Bmatrix}{{{\frac{\int{{\mathbb{d}^{3}x}\; ɛ\;{\chi^{(3)}\begin{bmatrix}{{{E_{1} \cdot E_{1}}}^{2} + {2{{E_{1} \cdot E_{1}^{*}}}^{2}} +} \\{{2\left( {E_{1} \cdot E_{1}^{*}} \right)\left( {E_{3} \cdot E_{3}^{*}} \right)} +}\end{bmatrix}}}}{\int{{\mathbb{d}^{3}x}\; ɛ\;\chi^{(3)}{E_{1}}^{2}}}\ldots} +}\;} \\\frac{\int{{\mathbb{d}^{3}x}\; ɛ\;{\chi^{(3)}\begin{bmatrix}{{2{{E_{1} \cdot E_{3}}}^{2}} + {2{{E_{1} \cdot E_{3}^{*}}}^{2}} +} \\{{3\left( {E_{1} \cdot E_{1}^{*}} \right)\left( {E_{1} \cdot E_{3}^{*}} \right)} +}\end{bmatrix}}}}{\int{{\mathbb{d}^{3}x}\; ɛ\;\chi^{(3)}{E_{1}}^{2}}}\end{Bmatrix}}} & (10) \\{\frac{\delta\;\omega_{3}}{\omega_{3}} = {{- \frac{1}{8}}\begin{Bmatrix}{{{\frac{\int{{\mathbb{d}^{3}x}\; ɛ\;{\chi^{(3)}\begin{bmatrix}{{{E_{3} \cdot E_{3}}}^{2} + {2{{E_{3} \cdot E_{3}^{*}}}^{2}} +} \\{{2\left( {E_{3} \cdot E_{3}^{*}} \right)\left( {E_{1} \cdot E_{1}^{*}} \right)} +}\end{bmatrix}}}}{\int{{\mathbb{d}^{3}x}\; ɛ\;\chi^{(3)}{E_{3}}^{2}}}\ldots} +}\;} \\\frac{\int{{\mathbb{d}^{3}x}\; ɛ\;{\chi^{(3)}\begin{bmatrix}{{2{{E_{1} \cdot E_{3}}}^{2}} + {2{{E_{3} \cdot E_{1}^{*}}}^{2}} +} \\{{\left( {E_{1}^{*} \cdot E_{1}^{*}} \right)\left( {E_{1}^{*} \cdot E_{3}} \right)} +}\end{bmatrix}}}}{\int{{\mathbb{d}^{3}x}\; ɛ\;\chi^{(3)}{E_{3}}^{2}}}\end{Bmatrix}}} & (11)\end{matrix}$

There is a subtlety in the application of perturbation theory todecaying modes, such as those of a cavity coupled to output ports. Inthis case, the modes are not truly eigenmodes, but are rather “leakymodes, and are not normalizable. Perturbative methods in this contextare discussed in more detail by, but for a tightly confined cavity modeit is sufficient to simply ignore the small radiating field far awayfrom the cavity. The field in the cavity is very nearly that of a trueeigenmode of an isolated cavity.

As stated above, one can arrive at the coupling coefficients by settingω_(k)→ω_(k)+δω_(k) in Eq. 1. However, the frequency perturbations δω_(k)are time-independent quantities, and we need to connect them to thetime-dependent a_(k) amplitudes. Therefore, to re-introduce the timedependence, one can use the slowly varying envelope approximation: aslowly varying, time-dependent amplitude a_(k)(t) is introduced into theunperturbed fields E_(k)→E_(k)a_(k)(t). The eigenmode must be normalizedso that |a_(k)|² is the energy, as assumed for the coupled-mode theory.Thus, we divide each E_(k) by

$\sqrt{\frac{1}{2}{\int ɛ}}{E_{k}}^{2}$

First, the χ⁽²⁾ medium is considered. Carrying out the abovesubstitutions in Eqs. 8-9 and grouping terms proportional a_(k) yieldsEqs. 3-4 with α_(ij) and β_(i) given by:

$\begin{matrix}{\alpha_{ij} = 0} & (12) \\{\beta_{1} = {\frac{1}{4}\frac{\int{{\mathbb{d}^{3}x}{\sum\limits_{ijk}\;{ɛ\;{\chi_{ijk}^{(2)}\left\lbrack {E_{1i}^{*}\left( {{E_{2j}E_{1k}^{*}} + {E_{1j}^{*}E_{2k}}} \right)} \right\rbrack}}}}}{\lbrack\int\rbrack{\mathbb{d}^{3}x}\; ɛ{{E_{1}}^{2} \cdot \left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{2}}^{2}}} \right\rbrack^{1/2}}}}} & (13) \\{\beta_{1} = {\frac{1}{4}\frac{\int{{\mathbb{d}^{3}x}{\sum\limits_{ijk}\;{ɛ\;\chi_{ijk}^{(2)}E_{2i}^{*}E_{1j}E_{1k}}}}}{\lbrack\int\rbrack{\mathbb{d}^{3}x}\; ɛ{{E_{1}}^{2} \cdot \left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{2}}^{2}}} \right\rbrack^{1/2}}}}} & (14)\end{matrix}$

A similar calculation yields the χ⁽³⁾ coupled-mode equations withcoefficients given by:

$\begin{matrix}{a_{ii} = {\frac{1}{8}\frac{{\int{{\mathbb{d}^{3}x}\;{{ɛ\chi}^{(3)} \cdot {{E_{i} \cdot E_{i}}}^{2}}}} + {{E_{i} \cdot E_{i}^{*}}}^{2}}{\left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{i}}^{2}}} \right\rbrack^{2}}}} & (15) \\{a_{ij} = {\frac{1}{4}\frac{{\int{{\mathbb{d}^{3}x}\;{{ɛ\chi}^{(3)} \cdot {E_{i}}^{2} \cdot {E_{2}}^{2}}}} + {{E_{1} \cdot E_{3}}}^{2} + {{E_{1} \cdot E_{3}^{*}}}^{2}}{\left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{i}}^{2}}} \right\rbrack^{2}}}} & (16) \\{{\alpha_{ij} = \alpha_{ji}}{\beta_{1} = {\frac{3}{8}\frac{\int{{\mathbb{d}^{3}x}\;{{ɛ\chi}^{(3)} \cdot \left( {E_{1}^{*} \cdot E_{1}^{*}} \right)^{2}}\left( {E_{1}^{*} \cdot E_{3}} \right)^{2}}}{\left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{1}}^{2}}} \right\rbrack^{3/2} \cdot \left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{3}}^{2}}} \right\rbrack^{1/2}}}}} & (17) \\{\beta_{3} = {\frac{1}{8}\frac{\int{{\mathbb{d}^{3}x}\;{{ɛ\chi}^{(3)} \cdot \left( {E_{1} \cdot E_{1}} \right)^{2}}\left( {E_{1} \cdot E_{3}^{*}} \right)^{2}}}{\left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{1}}^{2}}} \right\rbrack^{3/2} \cdot \left\lbrack {\int{{\mathbb{d}^{3}x}\; ɛ{E_{3}}^{2}}} \right\rbrack^{1/2}}}} & (18)\end{matrix}$

Note that Eqs. 13-15 verify the conditions ω₁β₁=ω₂β*₂ and ω₁β₁=ω₃β*₃,previously derived from conservation of energy—for χ⁽²⁾, this requiresthat one apply the symmetries of the χ_(ijk) ⁽²⁾ tensor, which isinvariant under permutations of ijk for a frequency-independent χ⁽²⁾.Furthermore, one can relate the coefficients α and β to an effectivemodal volume V. In particular, the strongest possible nonlinear couplingwill occur if the eigenfields are a constant in the nonlinear materialand zero elsewhere. In this case, any integral over the fields willsimply yield the geometric volume V of the nonlinear material. Thus, forthe χ⁽²⁾ effect one would obtain β_(i)˜χ⁽²⁾/√{square root over (V∈)};similarly, for the χ⁽³⁾ effect one can obtain α_(ij), β_(i)˜χ⁽³⁾/V∈.This proportionality to 1/√{square root over (V)} and 1/V carries overto more realistic field profiles, and in fact could be used to define amodal volume for these effects.

To check the predictions of the χ⁽³⁾ coupled-mode equations, a FDTDsimulation is performed of the one-dimensional waveguide-cavity systemshown in FIG. 1B, whose analytical properties are uniquely suited tothird-harmonic generation. This geometry consists of a semi-infinitephotonic-crystal structure made of alternating layers of dielectric(∈₁=13 and ∈₂=1) with period a and thicknesses given by the quarter-wavecondition (d₁=√{square root over (∈₂)}/(√{square root over(∈₁)}+√{square root over (∈₂)}) and d₂=a−d₁, respectively). Such aquarter-wave stack possesses a periodic sequence of photonic band gapscentered on frequencies ω₁=(√{square root over (∈₁)}+√{square root over(∈₂)}/4√{square root over (∈₁∈₂)} (in units of 2πc/a) for the lowestgap, and higher-order gaps centered on odd multiples of ω₁.

Moreover, a defect formed by doubling the thickness of a ∈₁ layercreates cavity modes at exactly the middle of every one of these gaps.Therefore, it automatically satisfies the frequency-matching conditionfor third-harmonic generation. In fact, it is too good: there will alsobe “ninth harmonic” generation from ω₃ to ω₉. This unwanted process isremoved, however, by the discretization error of the FDTD simulation,which introduces numerical dispersion that shifts the higher-frequencymodes. To ensure the ω₃=3ω₁ condition in the face of this dispersion,the structure is slightly perturbed increasing the dielectric constantslightly at the nodes of the third-harmonic eigenfield to tune thefrequencies. The simulated crystal was effectively semi-infinite, withmany more layers on the right than on the left of the cavity. On theleft of the cavity, after two period of the crystal the material issimply air (∈=1), terminated by a perfectly matched layer (PML)absorbing boundary region.

The cavity is excited with an incident plane wave of frequency ω₁, andcompute the resulting reflection spectrum. The reflected power at ω₃,the third-harmonic generation, was then compared with the prediction ofthe coupled-mode theory. The frequencies, decay rates, and α and βcoefficients in the coupled-mode theory were computed from a linear FDTDsimulation in which the eigenmodes were excited by narrow-band pulses.The freely available FDTD code of was employed.

The results are shown in FIG. 2, in which the output power at ω₁ andω₃=3ω₁ is denoted by |s¹⁻|² and |s³⁻|², respectively, while the inputpower at ω₁ is denoted by |s₁₊|². In particular, we plot convenientdimensionless quantities: the third-harmonic conversion efficiency|s³⁻|²/|s₁₊|² as a function of the dimensionless product n₂|s₁₊|² interms of the standard Kerr coefficient n₂=3ω⁽³⁾/4c∈. There is clearagreement between the FDTD and CMT for small values of n₂|s₁₊|² (inwhich limit the conversion goes quadratically with n₂|s₁₊|²). However,as the input power increases, they eventually begin to disagree, markingthe point where second-order corrections are required. This disagreementis not a practical concern, however, because the onset of second-ordereffects coincides with the limits of typical materials, which usuallybreak down for Δn/n≡χ⁽³⁾ max |E|²/2∈>1%. This is the maximum index shiftΔn/n is plotted in FIG. 2.

Also shown in FIG. 2 is a plot of Δω₁/ω₁=Re[δω₁/ω₁]. As expected, whenΔω₁ is of the order of 1/Q₁˜10⁻³, the frequency shift begins to destroythe frequency matching condition, substantially degrading thethird-harmonic conversion. It might seem that Δn/n and Δω₁/ω₁ should becomparable, but this is not the case because Δn/n is the maximum indexshift while Δω₁/ω₁ is due to an average index shift.

More specifically, the details of our simulation are as follows. Tosimulate a continuous wave (CW) source spectrum in FDTD, one can employa narrow-frequency Gaussian pulse incident from the air region. Thispulse is carefully normalized so that the peak intensity is unity, tomatch the CMT. The field in the air region is Fourier transformed andsubtracted from the incident field to yield the reflected flux. Usingonly two periods of quarter-wave stack on the left of the cavity weobtained two cavity modes with real frequencies ω₁=0.31818 (2πc/a),ω₂=0.95454 (2πc/a) and quality factors Q₁=1286 and Q₃=3726,respectively. Given these field patterns, one can compute the α_(ij) andβ_(i) coefficients. The following coupling coefficients are obtained, inunits of χ⁽³⁾: α₁₁=4.7531×10⁻⁴, α₂₂=5.3306×10⁻⁴, α₁₂=α₂₁=2.7847×10⁻⁴,β₁=(4.55985−0.7244)×10⁻⁵.

The conditions under which one may achieve complete frequency conversionis being considered: 100% of the incident power converted to output atthe second or third harmonic frequency. As we shall see, this is easiestto achieve in the χ⁽²⁾ case, and requires additional design criteria inthe χ⁽³⁾ case.

The key fact in a χ⁽²⁾ medium is that there are no frequency-shiftingterms (α=0), so the resonance condition ω₂=2ω₁ is not spoiled as oneincreases the power. The only requirement that we must impose is thatexternal losses such as absorption are negligible (τ_(k,e)>>τ_(k,s)). Inthis case, 100% conversion corresponds to setting s¹⁻=0 in thesteady-state. Using this fact, an input source s₊(t)=s₁₊exp(iw₁t) yieldsthe following condition on the input power for 100% conversion:

$\begin{matrix}{{s_{1 +}}^{2} = {\frac{2}{\omega_{1}^{2}{\beta_{1}}^{2}\tau_{2,s}\tau_{1,s}^{2}} = \frac{\omega_{1}}{2{\beta_{1}}^{2}Q_{2}Q_{1}^{2}}}} & (19)\end{matrix}$

A similar dependence of efficiency on Q₁ ²Q₂ was previously observedalthough a critical power was not identified. Thus, we can always choosean input power to obtain 100% conversion. If Q₁˜Q₂, then this criticalpower scales as V/Q³ where V is the modal volume, recall thatβ˜1/√{square root over (V)}.

This is limited, however, by first-order approximation: if the inputpower becomes so large that second-order effects (or material breakdown)become significant, then this prediction of 100% conversion is no longervalid. However, if one chooses Q₁ and/or Q₂ to be sufficiently large,then the critical power can be made arbitrarily small in principle. Notonly does the critical power decrease with Q³, but the field intensityin the cavity (|a_(i)|²) decreases as V/Q₁Q₂, and thus one can avoidmaterial breakdown as well as lowering the power.

To illustrate second-harmonic conversion for a χ⁽²⁾, medium, one canplot the solution to the coupled-mode equations as a function of inputpower in FIG. 3. The 100% conversion at the predicted critical power isclearly visible. For this calculation, one can choose modal parameterssimilar to the ones from the FDTD computation before: ω₁=0.3, χ₂=0.6,Q₁=10⁴, Q₂=2×10⁴, with dimensionless β₁=(4.55985−0.7244)×10⁻⁵.

A χ⁽³⁾ medium, on the other hand, does suffer from nonlinear frequencyshifts. For example, FIG. 2, which is by no means the optimal geometry,exhibits a maximal efficiency of |s³⁻|²/s₁₊|²≈4×10⁻³, almost threeorders of magnitude away from complete frequency conversion. On theother hand, one can again achieve 100% conversion if one can forceα_(ij)=0, which can be done in two ways. First, one could employ twoχ⁽³⁾ materials with opposite-sign χ⁽³⁾ values.

For example, if the χ⁽³⁾ is an odd function around the cavity center,then the integrals for α_(ij) will vanish while the β integrals willnot. (In practice, α<<β should suffice.) Second, one couldpre-compensate for the nonlinear frequency shifts: design the cavity sothat the shifted frequencies, at the critical power below, satisfy theresonant condition ω₃+Δω₃=3(ω₁+Δω₁). Equivalently, design the device forα_(ij)=0 and then adjust the linear cavity frequencies a posteriori tocompensate for the frequency shift at the critical power.

If α_(ij) is thereby forced to be zero, and we can also neglect externallosses (absorption, etc.) as above, then 100% third-harmonic conversion(s¹⁻=0) is obtained when:

If Q₁˜Q₃, then this critical power scales as V/Q² where V is the modalvolume (recall that β˜1/V). This is precisely the scaling that waspredicted for the power to obtain nonlinear bistability in a single-modecavity. Similarly, one finds that the energy density in the cavity(|a_(i)|²) decreases proportional to V/√{square root over (Q₁Q₃)}.

$\begin{matrix}{{s_{1 +}}^{2} = {\left\lbrack \frac{4}{3\omega_{1}^{2}{\beta_{1}}^{2}\tau_{1,s}^{3}\tau_{3,s}} \right\rbrack^{1/2} = \left\lbrack \frac{\omega_{1}\omega_{3}}{12{\beta_{1}}^{2}Q_{1}^{3}Q_{3}} \right\rbrack^{1/2}}} & (20)\end{matrix}$If Q₁˜Q₃, then this critical power scales as V/Q² where V is the modalvolume (recall that β˜1/V). This is precisely the scaling that waspredicted for the power to obtain nonlinear bistability in a single-modecavity. Similarly, one finds that the energy density in the cavity(|a_(i)|²) decreases proportional to V/√{square root over (Q₁Q₃)}.

It has been demonstrated that the third-harmonic conversion for α_(ij)=0by plotting the solution to the coupled-mode equations as a function ofinput power in FIG. 4. Again, 100% conversion is only obtained at asingle critical power. Here, one can use the same parameters as in theFDTD calculation, but with α=0. In this case, comparing with FIG. 2, thecomplete frequency conversion occurs at a power corresponding toΔn/n≈10⁻². This is close to the maximum power before coupled-mode theorybecomes invalid, either because of second-order effects or materialbreakdown, but one could easily decrease the critical power byincreasing Q.

For both the χ⁽²⁾ and the χ⁽³⁾ effects, in FIGS. 3-4, the harmonicconversion efficiency goes to zero if the input power (or χ) is eithertoo small or too large. It is not surprising that frequency conversiondecreases for low powers, but the decrease in efficiency for high powersis less intuitive. It corresponds to a well-known phenomenon incoupled-mode systems: in order to get 100% transmission from an inputport to an output port, the coupling rates to the two ports must bematched in order to cancel the back-reflected wave. In the present case,the coupling rate to the input port is ˜1/Q₁, and the coupling rate tothe output “port” (the harmonic frequency) is determined by the strengthof the nonlinear coupling. If the nonlinear coupling is either too smallor too large, then the rates are not matched and the light is reflectedinstead of converted. On the other hand, for large input powers, whilethe conversion efficiency as a fraction of input power goes to zero, theabsolute converted power (|s²⁻|² or |s³⁻|²) goes to a constant.

In practice, a real device will have some additional losses, such aslinear or nonlinear absorption and radiative scattering. Such losseswill lower the peak conversion efficiency below 100%. As we show in thissection, their quantitative effect depends on the ratio of the loss rateto the total loss rate 1/Q. We also solve for the critical input powerto achieve maximal conversion efficiency in the presence of losses.

For a χ⁽²⁾ medium with a linear loss rate 1/τ_(k,e), we solve Eqs. 3-4for |s²⁻|² and enforce the condition for maximal conversion efficiency:d/dt(|s²⁻|²/|s₁₊|²)=0. Thus, the following optimal input power andconversion efficiency is obtained: It immediately follows that for zeroexternal losses, i.e. τ_(k)=τ_(k,s), Eq. 22 gives 100% conversion andEq. 21 reduces to Eq. 19. For small external losses τ_(k,s)<<τ_(k,e),the optimal efficiency is reduced by the ratio of the loss rates, tofirst order:

$\begin{matrix}{{s_{1 +}}^{2} = \frac{2\tau_{1,s}}{\omega_{1}^{2}{\beta_{1}}^{2}\tau_{1}^{3}\tau_{2}}} & (21) \\{\frac{{s_{2 -}}^{2}}{{s_{1 +}}^{2}} = \frac{\tau_{1}\tau_{2}}{\tau_{1,s}\tau_{2,s}}} & (22)\end{matrix}$It immediately follows that for zero external losses, i.e.τ_(k)=τ_(k,s), Eq. 22 gives 100% conversion and Eq. 21 reduces to Eq.19. For small external losses τ_(k,s)<<τ_(k,e), the optimal efficiencyis reduced by the ratio of the loss rates, to first order:

$\begin{matrix}{\frac{{s_{2 -}}^{2}}{{s_{1 +}}^{2}} \approx {1 - \left( {\frac{\tau_{2,s}}{\tau_{2,e}} + {\frac{\tau_{1,s}}{\tau_{1,e}}.}} \right)}} & (23)\end{matrix}$A similar transmission reduction occurs in coupled-mode theory when anysort of loss is introduced into a resonant coupling processThe same analysis for χ⁽³⁾ yields the following critical input power andoptimal efficiency:

$\begin{matrix}{{s_{1 +}}^{2} = \left\lbrack \frac{4\tau_{1,s}^{2}}{\omega_{1}^{2}{\beta_{1}}^{2}\tau_{1}^{5}\tau_{2}} \right\rbrack^{1/2}} & (24) \\{\frac{{s_{3 -}}^{2}}{{s_{1 +}}^{2}} = \frac{\tau_{1}\tau_{3}}{\tau_{1,s}\tau_{3,s}}} & (25)\end{matrix}$where by comparison with Eq. 22, a first-order expansion for low-lossyields an expression of the same form as Eq. 23: the efficiency isreduced by the ratio of the loss rates, with τ₂ replaced by τ₃.

A χ⁽³⁾ medium can also have a nonlinear “two-photon” absorption,corresponding to a complex-valued χ⁽³⁾, which gives an absorptioncoefficient proportional to the field intensity. This enters thecoupled-mode equations as a small imaginary part added to α, even if onesets the real part of α to zero. The corresponding effect on β is just aphase shift. That yields a nonlinear (NL) τ_(k,e) of the following form,to lowest order in the loss:

$\begin{matrix}{\frac{1}{\tau^{1,e}{NL}} \approx {\omega_{1}{{Im}\left\lbrack {{\alpha_{11}\frac{\tau_{1,s}}{2}{s_{1 +}}^{2}} + {\alpha_{13}\frac{\tau_{3,s}^{2}\tau_{1,s}^{3}}{8}\omega_{3}^{2}{\beta_{3}}^{2}{s_{1 +}}^{6}}} \right\rbrack}}} & (26) \\{\frac{1}{\tau^{3,e}{NL}} \approx {\omega_{3}{{Im}\left\lbrack {{\alpha_{31}\frac{\tau_{1,s}}{2}{s_{1 +}}^{2}} + {\alpha_{33}\frac{\tau_{3,s}^{2}\tau_{1,s}^{3}}{8}\omega_{3}^{2}{\beta_{3}}^{2}{{s_{1 +}}^{6}.}}} \right\rbrack}}} & (27)\end{matrix}$These loss rates can then be substituted in the expression for thelosses above, in which case one obtains the following optimal efficiencyof third-harmonic generation, to lowest-order, not including linearlosses: Thus, the nonlinear loss is proportional to the ratio Imα/|β|,which is proportional to Imχ⁽³⁾/|χ⁽³⁾|.

$\begin{matrix}{\frac{{s_{3 -}}^{2}}{{s_{1 +}}^{2}} \approx {1 - {\frac{\tau_{3,s}}{\beta_{1}}\sqrt{\frac{\tau_{3,s}}{\tau_{1,s}}}{{Im}\left\lbrack {\frac{\alpha_{11} + {3\alpha_{13}}}{\tau_{3,s}} + \frac{\alpha_{13} + {3\alpha_{33}}}{\tau_{1,s}}} \right\rbrack}}}} & (28)\end{matrix}$Thus, the nonlinear loss is proportional to the ratio Imα|β|, which isproportional to Imχ⁽³⁾/|χ⁽³⁾|.

The following are possible structures and/or devices that can be usedfor and/or make use of complete frequency conversion. The goal is to useEqs. 12-18 so as to maximize β and reduce α (in the case of χ⁽³⁾ media).One possible cavity structure is a ring resonator, such as the onesshown in FIGS. 5A-5B, which is essentially are dielectric waveguidesarranged into a loop. Such resonators can come in many forms, such asrings, disks, spheres, non-circular rings, etcetera, with many degreesof freedom, such as the ring inner and outer shapes, that can be used tooptimize the harmonic modes and their overlap to improve harmonicgeneration.

In particular, FIG. 5A shows a schematic diagram of ring resonator 20with uniform waveguide coupling to a waveguide structure 24 and FIG. 5Bshows a schematic diagram of a ring resonator 22 having an asymmetricalwaveguide coupling. In FIG. 5B, the left waveguide 26 allows forpropagating modes of frequency ω and lω, while the right waveguide 28does not.

Such resonators can be evanescently coupled to many sorts of waveguidesadjacent to the ring, either above or to the side, including opticalfibers as well as on-chip dielectric “strip” or “rib” waveguides.Ideally, these will be arranged so that the light from the cavitycouples primarily to a single output channel. This can be accomplishedin several ways. For example, we could employ an asymmetricalwaveguide-cavity geometry, as in FIG. 5B, where one end of the waveguideis terminated in some way, ideally by a mirror such a photonic crystal,such as a periodic structure with a band gap reflecting light back alongthe waveguide. Another approach could be to use both the right and leftsides of the waveguide as a simultaneous input and output port, whichcould be joined by a Y-coupler of some form.

A dielectric waveguide with a one-dimensional periodicity, for example,a periodic sequence of holes or a periodic grating along the side of thewaveguide can have a photonic band gap in its guided modes. This bandgap can be used to trap light in a cavity by making a defect in theperiodicity, and these cavity modes could be used for harmonicgeneration. Like the one-dimensional photonic crystal consideredearlier, a periodic dielectric waveguide can have higher-order band gapsthat can be used to confine the harmonic mode(s), and can also have bandgaps at different frequencies for different polarizations which couldalso be used to confine the harmonic modes. Such defect-cavity designshave numerous degrees of freedom in their geometry which can be used tooptimize the coupling between the fundamental and harmonic modes.

A 2D photonic-crystal slab geometry can also be used as a possibledevice. Such slabs can be used to create cavities that confine light inthe plane via a photonic band gap. They can be designed to supportmultiple cavity modes at harmonic frequencies by, for example, utilizinghigher-order band gaps or band gaps in different polarizations.

A practical and useful application of complete frequency conversion isthat of high frequency generation of light sources. By employing asystem, shown in FIG. 6, one can convert 100% of the light given off bya laser, led or other light source 32 to a higher harmonic. The angularoffset θ of the qw-stack cavity system 34 from the source is there toensure that one is dealing with a single port (channel) cavity and canbe tuned so as to force 100% of the outgoing light at lω to travel inthe x-direction parallel to the source thus ensuring complete frequencyconversion.

Yet another possibility is to have the source and the Bragg-mirror, orany two-channel cavity with two available modes, parallel to each other.In this case, the fact that a two-port cavity is provided will changethe maximal achievable efficiency. However, one can still enable 100\%conversion efficiency provided that the coupling to the two channels isasymmetrical. Specifically, one must design one of the ports to couplestrongly to the first-harmonic frequency ω₁ while suppressing couplingto the higher-order harmonic frequency ω₁, and design the second port soas to achieve the inverse effect. This enables one to describe thewaveguide-cavity, effectively, as a one-port channel for both thefundamental and higher harmonic frequencies.

For example, an asymmetrical waveguide-cavity structure that satisfiesthe conditions given above can be obtained by careful design of twoBragg-mirrors: one of which should support a band-gap at ω₁, and asmaller band-gap at ω₂ and the other with a similar (but inverted)structure, i.e. small band-gap at ω₁ and larger band-gap at ω₂.

An important nonlinear process which was neglected in the previousanalysis is that of sum-frequency generation, or the generation of lightwith frequency ω₁+ω₂ from two input signals of frequencies ω₁ and ω₁.The existence of a critical input power for which one could achieve 100%frequency conversion, though not shown above, is definitively more thanfeasible based on similar arguments as above, i.e. rate matchingconditions. Such a device would require of a 3-mode cavity withfrequencies ω₁, ω₂ and ω₁+ω₂. This could be used to make very longwavelength sources.

FIG. 7 shows a schematic diagram of the geometry of a 3d photoniccrystal (PhC) cavity 40 created by adding a defect 42 in a rod-layer 44of a (111) fcc lattice of dielectric rods. Inset 46 shows a planar imageof the fundamental mode of the cavity 40. A complete gap system, such asthe PhC cavity 40 shown in FIG. 7, can also be used as a harmonicconverter with other cavity structures. Such a cavity would dramaticallyenhance the nonlinear interaction, reduce radiative and material losses,and thus potentially decrease the critical power by orders of magnitude.Since the Q of such a structure does not saturate but rather increasesexponentially with the number of surrounding layers, it would be highlydesirable as a low-power device. Again, higher-order bandgaps could beemployed to confine modes at harmonic frequencies.

In the case of second-harmonic generation, in order to preventsign-oscillations in the cavity modes from making the overlap integralsmall, a variety of techniques could be used, ranging from simpleoptimization of the cavity geometry to maximize the overlap, to usingnon-uniform “poling” of the materials so that χ⁽²⁾ is not uniform overthe cavity (for example, it could be concentrated in a particularregion, or even oscillate in sign matching the relative signs of thefundamental and harmonic fields).

The same principles apply to nonlinear frequency conversion in otherwave-propagation phenomena, such as acoustic waves, water waves, and soon.

The invention presents a rigorous coupled-mode theory for second- andthird-harmonic generation in doubly resonant nonlinear cavities,accurate to first order in the nonlinear susceptibility and validatedagainst a direct FDTD simulation. The invention predicts severalinteresting consequences. First, it is possible to design the cavity toyield 100% frequency conversion in a passive (gain-free) device, evenwhen nonlinear down-conversion processes are included, limited only byfabrication imperfections and losses. Second, this 100% conversionrequires a certain critical input power—powers either too large or toosmall lead to lower efficiency. Third, the invention describes how tocompensate for the self-phase modulation in a χ⁽³⁾ cavity. Themotivation for this invention was the hope that a doubly resonant cavitywould lead to 100% conversion at very low input powers.

A typical nonlinear material is gallium arsenide (GaAs), with χ⁽²⁾≈145pm/V and n₂=1.5×10⁻¹³ cm²/W at 1.5 μm. Al doping is usually employed todecrease nonlinear losses near resonance. Although this has both χ⁽²⁾and χ⁽³⁾ effects, one can selectively enhance one or the other bychoosing the cavity to have resonances at either the second or thirdharmonic. Many well confined optical cavity geometries are available atthese wavelengths and have been used for nonlinear devices, such as ringresonators or photonic-crystal slabs.

Conservative parameters are assumed for the cavity: a lifetime Q₁=1000,Q₂=2000, Q₃=3000, and a modal volume of 10 cubic half-wavelengths(V≈10(λ/2n)³) with roughly constant field amplitude in the nonlinearmaterial, worse than a realistic case of strongly peaked fields. In thiscase, the critical input power becomes approximately 20 mW forsecond-harmonic generation and 0.2W for third-harmonic generation with amoderate peak index shift Δn/n≈10⁻³, justifying the first-orderapproximation.

Using the expressions for α and β, optimized cavities for harmonicgeneration can be designed using standard methods to compute the lineareigenmodes. In practice, experimentally achieving cavity modes with“exactly” harmonic frequencies, matched to within the fractionalbandwidth 1/Q, is a challenge and may require some external tuningmechanism. For example, one could use the nonlinearity itself fortuning, via external illumination of the cavity with an intense “tuning”beam at some other frequency. Also, although one can directly integratethe coupled-mode equations in time, the invention intends to supplementthis with a linearized stability analysis at the critical power. This isparticularly important for the χ⁽³⁾ case, where pre-correcting thefrequency to compensate the nonlinear frequency shift (self-phasemodulation) may require some care to ensure a stable solution.

Although the present invention has been shown and described with respectto several preferred embodiments thereof, various changes, omissions andadditions to the form and detail thereof, may be made therein, withoutdeparting from the spirit and scope of the invention.

1. A doubly-resonant cavity structure comprising cavity structure havinga plurality of resonant modes so as to allow total frequency conversionfor second or third-harmonic generation using χ⁽²⁾ and χ⁽³⁾nonlinearities between said cavity structures, said total frequencyconversion is efficiently optimized by determining a critical powerallowing for such total frequency conversion to occur depending only onthe quality factors and the frequencies of said cavity structures. 2.The doubly-resonant cavity structure of claim 1 further comprisingcompensation for the self-phase modulation in a χ⁽³⁾ cavity structure.3. The doubly-resonant cavity structure of claim 1, wherein said atcavity structures comprises lifetime values between 1000 and
 3000. 4.The doubly-resonant cavity structure of claim 1, wherein said cavitystructures comprises a ring resonator coupled to a waveguide structure.5. The doubly-resonant cavity structure of claim 1, wherein said atleast one doubly-resonant cavity structures comprises a ring resonatorasymmetrically coupled to a waveguide structure.
 6. The doubly-resonantcavity structure of claim 1, wherein said cavity structures comprises aFabry-Perot cavity structure coupled to a quarter-wave stack.
 7. Thedoubly-resonant cavity structure of claim 1, wherein said cavitystructures comprises a led light source and quarter-wave stack.
 8. Thedoubly-resonant cavity structure of claim 1, wherein said cavitystructures comprises a photonic crystal cavity.
 9. The doubly-resonantcavity structure of claim 8, wherein said photonic crystal cavitystructure comprises a defect in a rod-layer.
 10. A method of performingtotal frequency conversion in a doubly-resonant cavity structurecomprising: providing cavity structures having a plurality of resonantmodes so as to allow said total frequency conversion for second orthird-harmonic generation using χ⁽²⁾ and χ⁽³⁾ nonlinearities betweensaid cavity structures; determining the cavity parameter of said cavitystructures; determining a critical power to efficiently optimized saidtotal frequency conversion using the cavity parameters of said cavitystructures; and applying said critical power so as to allow totalfrequency conversion between said cavity structures to occur.
 11. Themethod of claim 10 further comprising providing compensation for theself-phase modulation in a χ⁽³⁾ cavity structure.
 12. The method ofclaim 10, wherein said cavity structures comprises lifetime valuesbetween 1000 and
 3000. 13. The method of claim 10, wherein said cavitystructures comprises a ring resonator coupled to a waveguide structure.14. The method of claim 10, wherein said cavity structures comprises aring resonator asymmetrically coupled to a waveguide structure.
 15. Themethod of claim 10, wherein said cavity structures comprises aFabry-Perot cavity structure coupled to a quarter-wave stack.
 16. Themethod of claim 10, wherein said cavity structures comprises a led lightsource and quarter-wave stack.
 17. The method of claim 10, wherein saidcavity structures comprises a photonic crystal cavity.
 18. The method ofclaim 17, wherein said photonic crystal cavity structure comprises adefect in a rod-layer.
 19. A method of forming a doubly-resonant cavitystructure comprising: forming a cavity structures having a plurality ofmodes so as to allow total frequency conversion for second orthird-harmonic generation using χ⁽²⁾ and χ⁽³⁾ nonlinearities betweensaid cavity structures; and determining the cavity parameters of saidcavity structures so as to determine the critical power needed toperform said total frequency conversion.
 20. The method of claim 19further comprising providing compensation for the self-phase modulationin a χ⁽³⁾ cavity structure.
 21. The method of claim 19, wherein saidcavity structures comprises lifetime values between 1000 and
 3000. 22.The method of claim 10, wherein said cavity structures comprises a ringresonator coupled to a waveguide structure.
 23. The method of claim 19,wherein said cavity structures comprises a ring resonator asymmetricallycoupled to a waveguide structure.
 24. The method of claim 19, whereinsaid cavity structures comprises a Fabry-Perot cavity structure coupledto a quarter-wave stack.
 25. The method of claim 19, wherein said cavitystructures comprises a led light source and quarter-wave stack.
 26. Themethod of claim 19, wherein said cavity structures comprises a photoniccrystal cavity.
 27. The method of claim 26, wherein said photoniccrystal cavity structure comprises a defect in a rod-layer.